Larger nearly orthogonal sets over finite fields

For a field $F$ and integers d and k, a set $A\subseteq F^d$ is called k-nearly orthogonal if its members are non-self-orthogonal and every $k+1$ vectors of $A$ include an orthogonal pair. We prove that for every prime p there exists some $\delta =\delta \left(p\right)>0$, such that for every field $F$ of characteristic p and for all integers $k\ge 2$ and $d\ge k$, there exists a k-nearly orthogonal set of at least $d^{\delta \cdot k/\log k}$ vectors of $F^d$. The size of the set is optimal up to the $\log k$ term in the exponent. We further prove two extensions of this result. In the first, we provide a large set $A$ of non-self-orthogonal vectors of $F^d$ such that for every two subsets of $A$ of size $k+1$ each, some vector of one of the subsets is orthogonal to some vector of the other. In the second extension, every $k+1$ vectors of the produced set $A$ include $\ell +1$ pairwise orthogonal vectors for an arbitrary fixed integer $1\le \ell \le k$. The proofs involve probabilistic and spectral arguments and the hypergraph container method.